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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{Curve Sketching}
\centerline{November 28, 2017}
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\noindent
{\it Background}
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When mathematicians and scientists want to understand graphs of functions and other curves
these days, they almost always turn to graphing calculators and mathematical software to gen-
erate the plots. So it might seem that learning to plot functions by hand is a waste of time. 
\emph{However}, in my experience, some familiarity with curve sketching by hand is a great 
way to solidify your understanding of the First and Second Derivative Tests, concavity, 
asymptotes, etc.  So while you may never need to do this again ``in real life,'' (that is, 
unless civilization collapses and we lose the technology to make calculators, batteries, 
generate electricity to run the Internet, etc.) 
it's still a worthwhile educational exercise.

Here's a checklist of things to do and look for, given $y = f(x)$ to plot:
\begin{enumerate}
\item Determine the domain of your function and note whether the graph has any vertical asymp-
totes.  Draw those in if there are any.
\item Compute the derivative $f'(x)$ and find the critical points $x = c$, and 
the critical values $f(c)$.
\item On each interval between adjacent critical points, determine the sign of $f'(x)$. 
Recall that intervals where $f'(x) > 0$ are intervals where $f$ is increasing and intervals 
where $f'(x) < 0$ are intervals where $f$ is decreasing.
\item Compute $f''(x)$ and determine any $x$ where $f''(x) = 0$ or where $f''(x)$ does not exist.
\item Determine the intervals where $f$ is concave up and concave down using the sign of 
$f''$.  Recall that \emph{inflection points} are places where the \emph{concavity changes} --
from up to down or vice versa. If your function has inflection
points, find them and the function values there.
\item Determine horizontal asymptotes (if any), by taking $\displaystyle{\lim_{x\to\infty} f(x)}$.
\item The critical points and the inflection points can be thought of as 
``transition points'' as described in Section 4.6 of the text.
\item You should now be prepared to sketch the graph using the transition points, 
the information about where the function is increasing, decreasing, concave up, 
and concave down. See Figures 1 and 2 on page 231 for the four basic local shapes 
of graphs and an example of how they can be put together.
\end{enumerate}
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\noindent
{\it Questions}
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Find the transition points (critical points and points of inflection), 
intervals of increase and decrease, intervals of upward and downward concavity, 
asymptotes, and sketch the graph:
\begin{enumerate}
\item[1.] $y = x^5 - 15x^3$ (Hint: This one has three critical points, three points of 
inflection, and no asymptotes.)
\item[2.] $y = x^3 \ln(x)$ (Hint: $\displaystyle{\lim_{x\to 0+} x^3\ln(x) = 0}$.)
\item[3.] 
$$y = \frac{x+1}{x^2 - 2x}$$
(Hint: This one has both horizontal and vertical asymptotes. Be sure to simplify
$f'(x)$ before computing the second derivative.)
\end{enumerate}
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